Another proof of a theorem of J. Beck

Let x be a real number, α an irrational number with continued fraction expansion [a0; a1 , . . .] and convergents pn/qn, N a positive integer, m ∈ ℤ+ be chosen such that qm ≤ N < qm+1, {x} = x - [x] the fractional part of x and TN(α) = ∑Nn=1∑nk=1({kα} - 1/2). J. BECK [1] proved that TN(α) - N/12 ∑mi=1(-1)iai = O(Nmax{ai|1 ≤i≤m + 1}). We give a shorter proof, thereby using Dedekind sums. We may (and do) assume w.l.o.g. that 0 < α < 1. There are non-negative integers b0, . . . , bm, such that 0≤bi≤ai+1, b0 < a1, bi = ai+1 ⇒ bi-1 = 0 and N = ∑mi=1biqi (the so-called Ostrowski-expansion of N with respect to α). Let us put bi = 0 for i > m and, for k≥0, Nk = ∑ki=0biqi. Then Nk < qk+1. Let us finally put B(x) = {x} - 1/2 and, for p, q ∈ ℤ, q > 0 and gcd(p, q) = 1, ∑q-1i=1B(i/q)B(pi/q) =: S(p, q).